An underlying asset with price process \{S_t\}_{t \in J} , where J = \{0 \Delta t, 2\Delta t, \dots, T\} as both a European put and a European call option contract with the same strike price K and maturity T written on it. At time t = 0 the price of the call option is given by,

<br />
\Pi_C (0) = e^{-rT} \sum_{j=0}^n {n \choose{j}} (p^*)^j (1-p^*)^{n-j}(S_0 u^j d^{n-j} - K)^+<br />

where p^* is the risk neutral probability. Write down the pricing formula \Pi_P (0) for the put option.
Derive the Put-Call parity equation,

<br />
\Pi_C (0) + Ke^{-rT} = \Pi_P (0) + S_0<br />

Solution...

the price \Pi_C(0) of a European call written on a non-dividend-paying stock is given by...

<br />
\Pi_C (0) = e^{-rT} \sum_{j=a_n}^n {n \choose{j}} (p^*)^j  (1-p^*)^{n-j}(S_0 u^j d^{n-j} - K)^+<br />

where a_n = \min \{j \in \mathbb{N}_0 \textrm{ such that } S_0u^jd^{n-j} \geq K \} and the price \Pi_P (0) of a the corresponding European put is given by

<br />
 \Pi_C (0) = e^{-rT} \sum_{j=0}^{a_n-1} {n \choose{j}} (p^*)^j  (1-p^*)^{n-j}(K-S_0 u^j d^{n-j})^+<br />

Hence for the put-call parity we get...

\Pi_C (0) - \Pi_P (0) = e^{-rT} \sum_{j=0}^n {n \choose{j}} (p^*)^j  (1-p^*)^{n-j}(S_0 u^j  d^{n-j} - K)
= S_0 \sum_{j=0}^n {n \choose{j}} (p^*u)^j  (1-p^*d)^{n-j} \frac{1}{j!} \frac{1}{(n-j)!} e^{-r \Delta t} + Ke^{-rT} \sum_{j=0}^n {n \choose{j}} (p^*)^j  (1-p^*)^{n-j}

= ...?

I need to simplify the \frac{1}{j!} \frac{1}{(n-j)!} e^{-r \Delta t} part so I can bring it into the (p^*u)^j  (1-p^*d)^{n-j} parts and make a substitution but I don't know how to go about doing this...